Contextual Applications of Differentiation
What you must understand
Derivatives model rates: velocity is the derivative of position, acceleration the derivative of velocity. Interpret the sign and magnitude of a rate in context (increasing/decreasing, speeding up/slowing down).
Related rates: identify quantities, write an equation linking them, differentiate with respect to time, and substitute known values. Always include $\frac{d}{dt}$ factors. Common setups involve expanding spheres, ladders, and cones.
Linear approximation and differentials estimate function values: $f(a+h)\approx f(a)+f'(a)h$. Understand local linearity.
L'Hôpital's Rule evaluates indeterminate forms $\frac{0}{0}$ and $\frac{\infty}{\infty}$ by differentiating numerator and denominator; verify the form before applying. Recognize other indeterminate forms and rewrite them.
Traps
- Applying L'Hôpital's Rule when the form is not indeterminate.
- Forgetting to differentiate every related-rates quantity with respect to time.
- Confusing units or sign conventions (negative velocity vs. decreasing speed).
- Mixing up which quantity is constant in related rates.
Worked example
A balloon's radius grows at $2$ cm/s. Volume $V=\frac{4}{3}\pi r^3$. Then $\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}$. At $r=5$: $\frac{dV}{dt}=4\pi(25)(2)=200\pi$ cm³/s.
Rule of thumb: before L'Hôpital, confirm the indeterminate form; in related rates, differentiate first, substitute second.